TRIPOLI, Libya — The Obama administration is confronting a legal and policy dilemma that could reshape how it pursues terrorism suspects around the world as investigators try to determine who was responsible for the Sept. 11 attack on the U.S. mission in Benghazi.

Should it rely on the FBI, treating the assaults on the two U.S. compounds like a regular crime for prosecution in U.S. courts? Can it depend on the dysfunctional Libyan government to take action? Or should it embrace a military option by ordering a drone strike — or sending more prisoners to Guantanamo Bay?

President Obama has vowed to “bring to justice” the killers of Ambassador J. Christopher Stevens and three other Americans. But nearly one month later, the White House has not spelled out how it plans to do so, even if it is able to identify and capture any suspects.

Each of the options is fraught with practical obstacles and political baggage. An unproductive, slow-moving investigation is complicating matters, with the FBI taking three weeks to reach the unsecured crime scene. Meanwhile, the administration has given contradictory assessments, initially suggesting the attack was committed in the heat of the moment by a mob and more recently saying it was planned by terrorists affiliated with al-Qaeda.

On Tuesday, Obama’s chief counterterrorism adviser, John O. Brennan, is scheduled to visit Tripoli to meet with senior Libyan officials and give a high-level kick to the investigation.

The White House is not ruling out any option, an administration official said. The official, who spoke on the condition of anonymity to describe the evolving policy, said the involvement of the FBI at this stage should not be taken as evidence that the administration plans to prosecute any suspects in U.S. courts.

More broadly, it remains uncertain whether the White House will respond to the fatal assault on the Americans in Benghazi as a criminal act or an act of war, a critical legal distinction that has gone unresolved in Washington since the other Sept. 11 attacks, in 2001.

“It brings into sharp focus a number of issues that the government has been dealing with since the beginning of the so-called war on terror,” said Karen J. Greenberg, director of the Center on National Security at the Fordham University School of Law. “It clarifies so beautifully all of the hard issues we’ve had to confront over the last 11 years.”

All of the options available to the United States could have lasting consequences in Libya, where a transitional government is plagued by infighting and elected leaders have been unable to assume the full reins of power.

“There is very strong public opinion about the Americans coming here and running the investigation,” said Saad el-Shlmani, a ministry spokesman. Some top officials, he added, see the country’s sovereignty at stake.

But deferring to Libya’s fragile justice system — still warped after 42 years of undemocratic rule by Moammar Gaddafi — hardly presents an attractive choice for the administration.

As of last weekend, the Libyan government still had not secured the ruins of the primary U.S. compound in Benghazi, let alone interviewed many witnesses. Libyan courts can be chaotic places, especially in Benghazi. Lawyers say security issues can paralyze the system, which is only slowly starting to assume trappings of ordinary procedure in a country that does not yet have a constitution.

Courts are “functioning in Benghazi, but they’re partially functioning,” said Col. Mohammed Gweider, the head of the special courts and prison in Tripoli that handle high-level cases. “It’s the government weakness that’s being reflected in the court system.”

Asked whether the Libyan justice system could handle a prosecution related to the Benghazi attack, he said, “God willing, it can be ready” by the time any suspects are charged and put on trial.

Among U.S. officials, however, doubts are hardening.

Sen. Bob Corker (R-Tenn.), a member of the Senate Foreign Relations Committee who was in Tripoli on Monday to meet with Libyan officials, said a “lack of institutions” in post-revolutionary Libya is hampering efforts to jointly investigate the attack.

“I don’t think there’s been much coordination at all,” he said in an interview. “My sense is that almost everything the American government knows about the situation is what the American government has derived on their own.”

Asked if he had confidence that the perpetrators would be brought to justice, Corker replied: “Anybody who’s seen even a glimpse of this would have to say that it’s going to be very difficult.”

In a previously undisclosed development, Corker said U.S. investigators are examining video from security cameras at the primary Benghazi compound to help them piece together what happened on Sept. 11 and identify participants in the attack.

Despite the obstacles, John B. Bellinger III, a legal adviser to the White House and State Department under President George W. Bush, predicted that because of the circumstances of the case, the Obama administration would seek to bring any suspects to the United States to face trial in a civilian court. “I would tend to think that this administration — and frankly even the Bush administration or a Romney administration — would try hard to apply a criminal law enforcement approach if possible,” Bellinger said.

Even if the FBI is able to identify and locate the suspects, however, arresting them and transferring them to the United States could be difficult, given the lack of an extradition treaty with Libya.

Without an extradition treaty, the Libyans could apprehend the suspects themselves and hand them over to the United States outside a normal legal process — though some critics might paint such an arrangement as an extralegal rendition.

Regardless of the mechanism, bringing the suspects to the United States would ignite a whole separate debate over whether to prosecute them in the regular civilian courts or before a military commission.

Congress last year passed a bill that generally prescribes military commissions for terrorism suspects affiliated with al-Qaeda. But Bellinger predicted that the administration would nevertheless seek to prosecute the Libyan suspects in a civilian courtroom.

“Some Republicans might complain that if the killers were associated with al-Qaeda, they ought to be tried before a military commission,” he said. “But the law passed last year gives the president the option to try the suspects in the federal courts.”

Obama has not hesitated to order drone strikes in other countries, such as Yemen and Pakistan, where terrorism suspects have eluded the grasp of law enforcement agents. But such a course might come at a steep political cost in Libya, disrupting its emergence as a democratic nation and imperiling ties with Washington.

Some Libyans remember the 1986 airstrikes on Tripoli ordered by President Ronald Reagan in response to suspicions that Libya was responsible for the bombing of a West Berlin disco that killed two U.S. service members and injured 79 others.

“For Libya [drone strikes] would be a disaster. Libya is in a very fragile place,” said Shlmani, the Foreign Ministry spokesman. “Any unilateral action by any country, but especially by the United States, would really be damaging.”

Whitlock reported from Washington. Ayman Alkekly in Tripoli and Anne Gearan in Washington contributed to this report.

Michael Birnbaum is The Post’s Moscow bureau chief. He previously served as the Berlin correspondent and an education reporter.

Craig Whitlock covers the Pentagon and national security. He has reported for The Washington Post since 1998.

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